# 4.7 Graphs of linear inequalities  (Page 3/10)

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The boundary line shown is $2x+3y=6$ . Write the inequality shown by the graph.

The line $2x+3y=6$ is the boundary line. On one side of the line are the points with $2x+3y>6$ and on the other side of the line are the points with $2x+3y<6$ .

Let’s test the point $\left(0,0\right)$ and see which inequality describes its side of the boundary line.

At $\left(0,0\right)$ , which inequality is true:

$\begin{array}{ccccccccccc}\hfill 2x+3y& >\hfill & 6\hfill & & & \hfill \text{or}\hfill & & & \hfill 2x+3y& <\hfill & 6?\hfill \\ \hfill 2x+3y& >\hfill & 6\hfill & & & & & & \hfill 2x+3y& <\hfill & 6\hfill \\ \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{>}\hfill & 6\hfill & & & & & & \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{<}\hfill & 6\hfill \\ \hfill 0& >\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{False}\hfill & & & & & & \hfill 0& <\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{True}\hfill \end{array}$

So the side with $\left(0,0\right)$ is the side where $2x+3y<6$ .

(You may want to pick a point on the other side of the boundary line and check that $2x+3y>6$ .)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality $2x+3y<6$ .

Write the inequality shown by the shaded region in the graph with the boundary line $x-4y=8$ .

$x-4y\le 8$

Write the inequality shown by the shaded region in the graph with the boundary line $3x-y=6$ .

$3x-y\le 6$

## Graph linear inequalities

Now, we’re ready to put all this together to graph linear inequalities.

## How to graph linear inequalities

Graph the linear inequality     $y\ge \frac{3}{4}x-2$ .

## Solution

Graph the linear inequality $y\ge \frac{5}{2}x-4$ .

Graph the linear inequality $y<\frac{2}{3}x-5$ .

The steps we take to graph a linear inequality are summarized here.

## Graph a linear inequality.

1. Identify and graph the boundary line.
• If the inequality is $\le \text{or}\ge$ , the boundary line is solid.
• If the inequality is<or>, the boundary line is dashed.
2. Test a point that is not on the boundary line. Is it a solution of the inequality?
3. Shade in one side of the boundary line.
• If the test point is a solution, shade in the side that includes the point.
• If the test point is not a solution, shade in the opposite side.

Graph the linear inequality $x-2y<5$ .

## Solution

First we graph the boundary line $x-2y=5$ . The inequality is $<$ so we draw a dashed line.

Then we test a point. We’ll use $\left(0,0\right)$ again because it is easy to evaluate and it is not on the boundary line.

Is $\left(0,0\right)$ a solution of $x-2y<5$ ?

The point $\left(0,0\right)$ is a solution of $x-2y<5$ , so we shade in that side of the boundary line.

Graph the linear inequality $2x-3y\le 6$ .

Graph the linear inequality $2x-y>3$ .

What if the boundary line goes through the origin? Then we won’t be able to use $\left(0,0\right)$ as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Graph the linear inequality $y\le -4x$ .

## Solution

First we graph the boundary line $y=-4x$ . It is in slope–intercept form, with $m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=0$ . The inequality is $\le$ so we draw a solid line.

Now, we need a test point. We can see that the point $\left(1,0\right)$ is not on the boundary line.

Is $\left(1,0\right)$ a solution of $y\le -4x$ ?

The point $\left(1,0\right)$ is not a solution to $y\le -4x$ , so we shade in the opposite side of the boundary line. See [link] .

Graph the linear inequality $y>-3x$ .

Graph the linear inequality $y\ge -2x$ .

Some linear inequalities have only one variable. They may have an x but no y , or a y but no x . In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

$\begin{array}{cccc}x=a\hfill & & & \text{vertical line}\hfill \\ y=b\hfill & & & \text{horizontal line}\hfill \end{array}$

Graph the linear inequality $y>3$ .

## Solution

First we graph the boundary line $y=3$ . It is a horizontal line. The inequality is>so we draw a dashed line.

We test the point $\left(0,0\right)$ .

$\begin{array}{}\\ y>3\hfill \\ \\ 0\overline{)>}3\hfill \end{array}$

$\left(0,0\right)$ is not a solution to $y>3$ .

So we shade the side that does not include (0, 0).

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes